ENDG 319 Section #10 (Hypothesis Testing) Summary Sheet H0: ; known and given. Data are either normally distributed or you are invoking CLT (n 30) Compute critical value z where If H1: 0 Reject H0 if z z P(Z< -z) = Compute statistic Compute critical value z where If H1: 0 Reject H0 if z z x 0 P(Z> z) = z Compute critical value z/2 n Reject H0 if If H1: 0 where P(Z< -z/2) = /2 and z z 2 or z z 2 P(Z> z/2) = /2 H0: ; ; unknown but n 30 so you invoke CLT. This is a “large sample test for a single mean”; given. Compute critical value z where If H1: 0 Reject H0 if z z P(Z< -z) = Compute statistic Compute critical value z where If H1: 0 Reject H0 if z z x 0 P(Z> z) = z Compute critical value z/2 s n Reject H0 if If H1: 0 where P(Z< -z/2) = /2 and z z 2 or z z 2 P(Z> z/2) = /2 H0: ; unknown and data is normally distributed; cannot invoke CLT; given. Compute critical value t where If H1: 0 Reject H0 if t t P(T< -t) = with df = n-1 Compute statistic Compute critical value t where If H1: 0 Reject H0 if t t x 0 P(T> t) = with df = n-1 t Compute critical value t/2 s n Reject H0 if If H1: 0 where P(T< -t/2) = /2 and t t 2 or t t 2 P(T> t/2) = /2; df = n-1 H0: d; , known and given. Data are either normally distributed or you invoke CLT (n 30) Compute critical value z where If H1: 1 2 d0 Reject H0 if z z P(Z< -z) = Compute statistic Compute critical value z where If H1: 1 2 d 0 Reject H0 if z z x1 x2 d 0 P(Z> z) = z Compute critical value z/2 12 n1 22 n2 Reject H0 if If H1: 1 2 d 0 where P(Z< -z/2) = /2 and z z 2 or z z 2 P(Z> z/2) = /2 H0: d; , unknown, but n 30: use “large sample test”; given. Compute critical value z where Reject H0 if z z If H1: 1 2 d0 P(Z< -z) = Compute statistic Compute critical value z where If H1: 1 2 d 0 Reject H0 if z z x1 x 2 d 0 P(Z> z) = z Compute critical value z/2 s12 n1 s 22 n 2 Reject H0 if If H1: 1 2 d 0 where P(Z< -z/2) = /2 and z z 2 or z z 2 P(Z> z/2) = /2 H0: d; = and both are unknown; data are normally distributed; given. Compute critical value t where Compute statistic Reject H0 if t t If H1: 1 2 d0 P(T< -t) = with df=n1+n2-2 x1 x2 d 0 t Compute critical value t where If H1: 1 2 d 0 Reject H0 if t t s p 1 n1 1 n2 P(T> t) = with df=n1+n2-2 Compute critical value t/2 ( n 1)s12 ( n 2 1)s 22 Reject H0 if with s 2p 1 If H1: 1 2 d 0 where P(T< -t/2) = /2 and n1 n 2 2 t t 2 or t t 2 P(T> t/2) = /2; df=n1+n2-2 Dr. Sameh Nassar ENDG 319 Section #10 (Hypothesis Testing) Summary Sheet H0: d; ≠ and both are unknown; data are normally distributed; given. Compute critical value t where x 1 x 2 d 0 ; If H1: 1 2 d0 Reject H0 if t t t P(T< -t) = 2 2 s 1 n1 s 2 n 2 Compute critical value t where If H1: 1 2 d 0 Reject H0 if t t 2 P(T> t) = s12 n1 s22 n 2 with df Compute critical value t/2 2 2 Reject H0 if s12 n1 s22 n 2 If H1: 1 2 d 0 where P(T< -t/2) = /2 and t t 2 or t t 2 P(T> t/2) = /2 n1 1 n 2 1 H0: Dd; ≠ and both are unknown; data are normally distributed AND PAIRED; given. If H1: D d0 If H1: D d 0 If H1: D d0 Compute critical value t where P(T< -t) = , df = n-1 Compute critical value t where P(T> t) = , df = n-1 Compute critical value t/2 where P(T< -t/2) = /2 and P(T> t/2) = /2, df = n-1 Reject H0 if Compute statistic t d d0 sd n Reject H0 if t t t t Reject H0 if 2 or t t t t 2 Formulating Hypothesis Testing Problems Hypotheses about a random variable x are often formulated in terms of its distributional properties. Example, if property is a: Null hypothesis H0: a = a0 Alternative hypothesis H1: a < a0 || a > a0 || a ≠ a0 Objective of hypothesis testing is to decide whether or not to reject this hypothesis. Decision is based on estimator â of a: Reject H0: If observed estimate â lies in rejection region Ra0 (i.e. â Ra0) Do not reject H0: Otherwise (i.e. â Ra0) Select rejection region to obtain desired error properties: Type I Error probability α is called the test significance level. 1- is said to be the power of a test; and is the Type II Error probability Computing and choosing sample size Suppose that we wish to test the hypothesis (H0: ; H1: ) with a significance level when 2 is known. For a specific alternative, say =0+, the power of our test is: 1- = P[x>a when =0+, which is equivalent to: P Z z P Z z n Dr. Sameh Nassar z Hence : n z 2 2 2 or z n z 2 2 /2 2 ( two tailed) ENDG 319 Section #10 (Hypothesis Testing) Summary Sheet H 0: = If H1: 2 02 If H1: 2 02 If H1: 2 02 Compute critical value 21- where P(2 < 21-) = Compute critical value 2 where P(2 > 2) = Compute critical values 2/2 and 21-/2 where P(2<21-/2) = /2 and P(2 > 2/2) = /2 Reject H0 if 2 < 21- Compute statistic (n 1)s 2 2 02 with df n 1 Reject H0 if 2 > 2 Reject H0 if 2<21-/2 or 2 > 2/2 H 0: = If H1: ≠ If H1:> or < Compute critical values F(df1, df2) where P(F > F) = /2 Compute critical values F(df1, df2) where P(F > F) = Compute statistic 2 1 2 2 s if s12 s 22 s with df1 = n1 - 1 and df 2 = n 2 - 1 or s2 F 22 if s12 s 22 s1 with df1 = n 2 - 1 and df 2 = n1 - 1 F (Goodness-for-Fit Test) H0: observation x follows a specified distribution f(x) Compute statistic k o e 2 i H1: x does not follow the distribution f(x) 2 i ei i 1 with df k 1 Reject H0 if F > F(df1, df2) Reject H0 if F > F(df1, df2) Reject H0 if 2 > 2 Note: Expected frequencies ei must be ≥ 5 => collapse classes where necessary Dr. Sameh Nassar ENDG 319 Section #11 Summary Sheet Simple Linear Regression and Correlation